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Pulsar Line of Death
We can think about a pulsar as a battery that drives currents along magnetic field lines. The two “terminals” of the battery are the pole and the edge of the polar cap. The radius of the polar cap is l \approx R\sqrt{R/cP} , where R is the radius of the pulsar, c is the speed of light and P is the spin period. The electric field is given by E \approx B l / P c where B is the magnetic field on the surface of the pulsar. The voltage drop is \Delta V \approx lE \approx BR^3/P^2 c^2 . This is enough to accelerate electrons to a Lorentz factor \gamma \approx q \Delta V/m c^2 where m is the mass of the electron and q is the elementary charge. These particles move along the field lines and emit curvature radiation in the process. The curvature radius of the magnetic field lines at the polar cap is given by r_c \approx \sqrt{R c P} , and the frequency of the emitted radiation is f \approx \frac{c}{r_c} \gamma^3 (two factors of the Lorentz factor come from the fact that the radiation is beamed, and another one from the Lorentz boost). We proceed to determine the condition under which the curvature radiation can interact with the magnetic field to produce pairs. We consider the magnetic field as though it is composed from virtual photons, and the real photon interacts with a group of coherent virtual photons to produce a pair. We proceed to calculate the properties of this group of virtual photons, and treat them as though they are a single particle. The momentum of EM fields is given by p = q A/c , where A is the vector potential. This vector potential can be constructed from a magnetic field and a length scale, which, since we consider electron - proton pairs, is the electron Compton wavelength \lambda = \frac{\hbar}{m_e c} , and so A \approx B \lambda . The energy of the group of virtual photons is \varepsilon \approx p c \approx B \frac{q \hbar}{m_e c} . If the photon were coming from a random direction, then the condition to create an electron positron pair would be \varepsilon h f > m^2 c^4 . However, since the photon is almost parallel to the magnetic field lines, only the transverse component of the momentum is available to produce pairs. Hence, the condition for pair production is \varepsilon h f \theta_p > m^2 c^4 , where \theta_p is the opening angle of the polar cap. The magnetic field cannot be measured directly, but it can be inferred from observations by measuring the change in period \dot{P} . Assuming the radiated dipole energy comes at the expense of the spin rotation energy M \frac{R^2}{P^3} \dot{P} \approx \frac{B^2 R^4}{c^3} \left(\frac{R}{P^2}\right)^2 \Rightarrow B^2 \approx M \frac{c^3}{R^4} P \dot{P} where M is the mass of the pulsar. The condition for pair production becomes \dot{P} \approx 3 \cdot 10^{-15} \left(\frac{P}{1 \, \rm s}\right)^{5/2} \left(\frac{M}{M_{\oplus}}\right)^{-1} \left(\frac{R}{10 \, \rm km}\right)^{-1/2} In pulsar where this condition is violated radio emission is supposedly suppressed. This scaling law has been reproduced previous works. Category:Stellar structure